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Identical Particles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: March 04, 2017)

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Identical Particles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: March 04, 2017) Identical particles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: March 04, 2017) In classical mechanics, identical particles (such as electrons) do not lose their "individuality", despite the identity of their physical properties. In quantum mechanics, the situation is entirely different. Because of the Heisenberg's principle of uncertainty, the concept of the path of an electron ceases to have any meaning. Thus, there is in principle no possibility of separately following each of a number of similar particles and thereby distinguishing them. In quantum mechanics, identical particles entirely lose their individuality. The principle of the in-distinguishability of similar particles plays a fundamental part in the quantum theory of system composed of identical particles. Here we start to consider a system of only two particles. Because of the identity of the particles, the states of the system obtained from each other by interchanging the two particles must be completely equivalent physically. 1 System of particles 1 and 2 k' , k" k' k" k" k' a 1 2 b 1 2 Even though the two particles are indistinguishable, mathematically a and b are distinct kets for k' k" . In fact we have a b 0 . Suppose we make a measurement on the two particle system. k' : state of one particle and k" : state of the other particle. We do not know a priori whether the state ket is k' k" or k" k' or -for that matter- any a 1 2 b 1 2 linear combination of the two: ca a cb b . 2 Exchange degeneracy A specification of the eigenvalue of a complete set of observables does not completely determine the state ket. Mathematics of permutation symmetry: Pˆ Pˆ k' k" k" k' . 12 a 12 1 2 1 2 b Clearly ˆ ˆ ˆ 2 P21 P12 , and P12 1. ˆ Under P12 , particle 1 having k' becomes particle 1 having k" ; particle 2 having k" becomes particle 1 having k' . In other words, it has the effect of interchanging 1 and 2. ((Note)) Identical particles 1 9/3/2017 Matrix element of Pˆ Pˆ in terms of k' k" and k" k' 21 12 a 1 2 b 1 2 Pˆ Pˆ k' k" k" k' , 12 a 12 1 2 1 2 b Pˆ Pˆ k" k' k' k" . 12 b 12 1 2 1 2 a ˆ ˆ Matrix element of P21 P12 ˆ 0 1 P12 1 0 or a b a 0 1 b 1 0 ˆ Eigensystem[ P12 ] = 1 (symmetric): P12 symm symm 1 1 ( k' k" k" k' ) (1ˆ Pˆ ) k' k" symm 2 1 2 1 2 2 12 1 2 where the symmetrizer is defined by 1 Sˆ (1ˆ Pˆ ) 2 12 = -1 (anti-symmetric) P12 anti anti 1 1 ( k' k" k" k' ) (1ˆ Pˆ ) k' k" anti 2 1 2 1 2 2 12 1 2 where the antisymmetrizer is defined by 1 Aˆ (1ˆ Pˆ ) 2 12 Identical particles 2 9/3/2017 Our consideration can be extended to a system made up of many identical particles. A transposition is a permutation which simply exchange the role of two of the particles, without touching others. Pˆ k' k" ..... k i k i1 ..... k j ..... k' k" ..... k j k i1 ..... k i ..... ij 1 2 i i1 j 1 2 i i1 j ˆ ˆ ˆ The transposition operators Pij are Hermitian ( Pij Pij ) ˆ 2 ˆ Pij 1 ˆ ˆ So that Pij is an unitary operator. The allowed eigenvalues of Pij are ±1. It is important to note, however, that in general ˆ ˆ [Pij , Pkl ] 0. (i) The first example ˆ Now we consider a permutation operator P123 for Pˆ k' k" k"' k' k" k"' k"' k' k" 123 1 2 3 2 3 1 1 2 3 for the system of 3 identical particles ( k' k" k''' ) 1 2 3 123 P 123 231 This means replacement of 12, 2 3, 3 1. 123 123213 P P P 123 231 213231 12 13 Quantum mechanically this is not correct. The correct one is ˆ ˆ ˆ P123 P13 P12 (ii) The second example 123 12 3132 P P P 123 231 132231 23 12 or quantum mechanically ˆ ˆ ˆ P123 P12P23 (iii) The third example Identical particles 3 9/3/2017 123 123321 P P P 132 312 321312 13 12 or quantum mechanically ˆ ˆ ˆ P132 P12 P13 ((Proof)) Pˆ Pˆ k' k" k''' Pˆ k''' k" k' k" k''' k' 12 13 1 2 3 12 1 2 3 1 2 3 and Pˆ k' k" k''' k' k" k''' k" k''' k' 132 1 2 3 3 1 2 1 2 3 ˆ ˆ ˆ Therefore we have P132 P12 P13 . Any permutation operators can be broken into a product of transposition operators. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 P132 P23 P12 P13 P23 P12 P13 P23 P12 P23 ... The decomposition is not unique. However, for a given permutation, it can be shown that the parity of the number of transposition into which it can be broken down is always the same: it is called the parity of the permutation. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 P132 P23P12 P13P23 P12P13 P23P12P23 even permutation ˆ ˆ ˆ ˆ ˆ P123 P12 P23 P13 P12 : even permutation ˆ P23 odd permutation ˆ P12 odd permutation ˆ P31 odd permutation ((Note)) ˆ ˆ ˆ P132 P321 P213 3 Symmetrizer and antisymmetrizer Now we consider the two Hermitian operators ˆ 1 ˆ S P : symmetrizer N! Identical particles 4 9/3/2017 ˆ 1 ˆ A P : antisymmetrizer N! ˆ ˆ where =1 if P is an even permutation and = -1 if P is an odd permutation. Sˆ Sˆ and Aˆ Aˆ . ((Theorem-1)) ˆ If P 0 is an arbitrary permutation operator, we have ˆ ˆ ˆ ˆ ˆ P 0 S SP 0 S ˆ ˆ ˆ ˆ P 0 A AP 0 0 A (1) ((Proof)) This is due to the fact that ˆ ˆ ˆ P 0P P such that 0 or 2 0 0 Consequently ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ P 0S P 0P P S N! N! ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ P 0 A P 0 P 0 P 0 A N! N! From Eq.(1), we see the following theorem ((Theorem-2)) Sˆ 2 Sˆ Aˆ 2 Aˆ and AˆSˆ SˆAˆ 0 Identical particles 5 9/3/2017 ((Proof)) ˆ 2 1 ˆ ˆ 1 ˆ ˆ S P S S S N! N! ˆ 2 1 ˆ ˆ 1 2 ˆ ˆ A P A A A N! N! ˆ ˆ ˆ 1 ˆ ˆ 1 ˆ AS A P S S 0 N! N! since half the are equal to 1 and half the equal to -1. We also note that 1 2 1 1 1 N! N! Sˆ and Aˆ are therefore projectors. Their action on any ket of the state space yields a completely symmetric or completely antisymmetric ket. ˆ ˆ ˆ P 0S S Pˆ Aˆ Aˆ 0 0 ˆ S S ˆ A A ((Example)) For N = 3, 1 Sˆ [1ˆ Pˆ Pˆ Pˆ Pˆ Pˆ ] 6 12 23 31 123 132 and 1 Aˆ [1ˆ Pˆ Pˆ Pˆ Pˆ Pˆ ] 6 12 23 31 123 132 where ˆ ˆ ˆ ˆ ˆ ˆ P123 P12P23 , P132 P12P13 Identical particles 6 9/3/2017 1 Sˆ Aˆ (1ˆ Pˆ Pˆ ) 1ˆ 3 123 132 4 Symmetrization postulate The system containing N identical particles are either totally symmetrical under the interchange of any pair (boson), or totally antisymmetrical under the interchange of any pair (fermion). ˆ Pij N ,B N ,B , ˆ Pij N ,F N ,F , where N ,B is the eigenket of N identical boson systems and N ,F is the eigenket of N identical fermion systems. ((Note)) It is an empirical fact that a mixed symmetry does not occur. Even more remarkable is that there is a connection between the spin of a particle and the statistics obeyed by it: Half-integer spin particles are fermion, while integer-spin particles are bosons. 5 Pauli exclusion principle Wolfgang Ernst Pauli (April 25, 1900 – December 15, 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle," involving spin theory, underpinning the structure of matter and the whole of chemistry. http://en.wikipedia.org/wiki/Wolfgang_Pauli Electron is a fermion. No two electrons can occupy the same state. We discuss the framatic difference between fermions and bosons. Let us consider two particles. Each of which can occupy only two states k' and k" . For a system of two fermions, we have no choice Identical particles 7 9/3/2017 1 1 k' k' [ k' k" k' k" ] 1 2 1 2 2 1 k'' k" 2 2 1 2 For bosons, there are three states. k' k' , 1 2 k" k" , 1 2 1 ( k' k" k" k' ) . 2 1 2 1 2 In contrast, for “classical particles” satisfying Maxwell-Boltzmann (M-B) statitics with no restriction on symmetry, we have altogether four independent states. k' k" , k" k' , k' k' and k" k" 1 2 1 2 1 2 1 2 We see that in the fermion case, it is impossible for both particles to occupy the same state.
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